Airline Fares Depend on Unobserved Route and Time Characteristics

Two Sources of Omitted Variable Bias

\[ \begin{aligned} \log(\text{fare})_{it} &= \beta_0 + \beta_1\log(\text{distance})_i + \beta_2\text{competition}_{it} + e_{it} \\ e_{it} &= \gamma_i + \delta_t + u_{it} \end{aligned} \]

Controlling for Distance and Year Dummies Does Not Remove Route Heterogeneity

The Proposed Model Falls Short

Combining FD with Year Dummies Addresses Both Sources

Two Strategies for Two-Way Heterogeneity

(1) FD + year dummies, or (2) full LSDV with dummies for both units and time periods.

Industry-Wide Patent Growth Requires Flexible Time Effects

Year Dummies Capture Any Form of Common Time Trends

Year Dummy Coefficients Measure Growth Rates

Conditional Expectations Are The Tool to Interpret

\[\begin{align*} \mathbb{E}[\log(\text{patents})_{it} \mid t=2005] &= \beta_0 + \beta_1\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2006] &= (\beta_0 + \beta_2) + \beta_1\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2007] &= (\beta_0 + \beta_3) + \beta_1\log(\text{R\&D})_{it} \end{align*}\]

• Average log change in patents across all firms
• Geometric mean growth rate of patents in the industry, holding R&D constant
  • \(\beta_2\): 2005 to 2006
  • \(\beta_3\): 2005 to 2007

Year dummies measure growth rates between periods — not “the level in 2006 vs 2005.”

Interactions Allow the R&D Elasticity to Vary Over Time

Heterogeneous Elasticities

\[ \begin{align*} \log(\text{patents})_{it} &= \beta_0 + \beta_1\log(\text{R\&D})_{it} + \beta_2\mathbb{1}[t=2006] + \beta_3\mathbb{1}[t=2007] \\ &+ \beta_4(\log(\text{R\&D})_{it} \times \mathbb{1}[t=2006]) + \beta_5(\log(\text{R\&D})_{it} \times \mathbb{1}[t=2007]) \\ &+ e_{it} \end{align*} \]

Conditional means by year

\[\begin{align*} \mathbb{E}[\log(\text{patents})_{it} \mid t=2005] &= \beta_0 + \beta_1\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2006] &= (\beta_0 + \beta_2) + (\beta_1 + \beta_4)\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2007] &= (\beta_0 + \beta_3) + (\beta_1 + \beta_5)\log(\text{R\&D})_{it} \end{align*}\]

Level Shifts and Slope Shifts Are Separately Identified

Decomposing Differential Effects

No need to interpret \(\beta_4\) and \(\beta_5\) individually; the conditional means do the work.

Gender Wage Gaps Changed After the Mining Boom

Three Patterns from the Data

• Gender: Men earn a constant wage premium over women
• Time trend: Wages grow over time for both groups
• Structural break (2005): After the mining company arrives, the male premium widens

The Empirical Model Interacts Gender, Time, and Post-2005

Eight Coefficients for Four Groups

\[ \begin{align*} \log(\text{wages})_{it} &= \beta_0 + \beta_1\mathbb{1}[i\text{ is male}] + \beta_2 t + \beta_3\mathbb{1}[t \geq 2005] \\ &+ \beta_4(\mathbb{1}[i\text{ is male}] \times t) + \beta_5(\mathbb{1}[i\text{ is male}] \times \mathbb{1}[t \geq 2005]) \\ &+ \beta_6(t \times \mathbb{1}[t \geq 2005]) + \beta_7(t \times \mathbb{1}[t \geq 2005] \times \mathbb{1}[i\text{ is male}]) \\ &+ e_{it} \end{align*} \]

This model captures level differences, trends, and how both changed after 2005, separately for men and women.

Conditional Means: Pre-2005

Women and Men Before the Structural Break

Women before 2005 (base category)

\[ \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=0,t<2005] = \beta_0 + \beta_2 t \]

Men before 2005

\[ \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=1,t<2005] = (\beta_0 + \beta_1) + (\beta_2 + \beta_4)t \]

\(\beta_1\) shifts the intercept; \(\beta_4\) shifts the slope.

Conditional Means: Post-2005

Women and Men After the Structural Break

Women after 2005

\[ \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=0,\; t \geq 2005] = (\beta_0 + \beta_3) + (\beta_2 + \beta_6)t\]

Men after 2005

\[\begin{align*} \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=1,\; t \geq 2005] &= (\beta_0 + \beta_1 + \beta_3 + \beta_5) \\ &\quad + (\beta_2 + \beta_4 + \beta_6 + \beta_7)t \end{align*}\]

Each coefficient modifies either the intercept or slope for a specific group-period combination.

Taking Differences Isolates Each Coefficient’s Role

Condition on Group, Then Difference

Coefficient Signs Follow Directly from the Figure

Economic Interpretation

Panel Data Follows the Same Units Over Time

Definition and Structure

• \(y_{it}\), \(x_{it}\) for \(i = 1, \ldots, N\) and \(t = 1, \ldots, T\)
• Panel data: same units observed across multiple time periods
• Cross-section: one \(t\) only
• Repeated cross-section: same population, different individuals each period

Error decomposition

\[ e_{it} = \alpha_i + v_{it} \]

Panel Structure Enables Identification of Parameters

Identification vs Estimation

Example: cannot estimate returns to education via FD if education does not change over time.

Education Is Measured with Error in Both Periods

True vs Observed Variables

\[ \text{education}_{it} = \text{education}^{*}_{it} + e_{it} \]

Assumptions

• \(e_{it}\) uncorrelated with true education and other variables
• Education varies little over time for adults

Cross-Sectional Attenuation Bias Shrinks the Coefficient Towards Zero

The Baseline Problem

Population model

\[ \log(\text{wage})_i = \alpha + \beta\text{education}^{*}_i + \epsilon_i \]

Observed model substitutes \(\text{education}_i = \text{education}^{*}_i + e_i\)

Attenuation bias

\[ \text{plim}\;\hat\beta = \beta \cdot \frac{\text{Var}(\text{educ}^*)}{\text{Var}(\text{educ}^*) + \text{Var}(e)} \]

The ratio is less than 1, so the coefficient is biased towards zero (derived in Topic 6).

First-Differencing Increases the Measurement Error Variance

The Panel Data Paradox

FD of observed education

\[ \Delta\text{education}_i = \Delta\text{education}^{*}_{i} + (e_{i2} - e_{i1}) \]

If \(e_{i1}\) and \(e_{i2}\) are uncorrelated

\[ \text{Var}(e_{i2} - e_{i1}) = \text{Var}(e_{i1}) + \text{Var}(e_{i2}) \]

Roommate Nationality May Affect Student Grades

Self-Selection into Rooms Creates Endogeneity

Let \(\text{same}_{it} = \mathbb{1}[i\text{ has same-nationality roommate in } t]\)

\[ \text{grades}_{it} = \alpha + \beta\;\text{same}_{it} + e_{it} \]

• Exogeneity holds under random assignment of roommates
• If students choose roommates: an omitted equation determines room selection
• More outgoing students may prefer different nationalities and perform differently academically
• \(\text{Cov}(\text{same}_{it}, e_{it}) \neq 0\) arises from selection, not unobserved heterogeneity

Two Years of Data Require Roommate Changes

Panel Structure and Exogenous Mobility

Year 1

\(\text{grades}_{i1} = \alpha + \beta\;\text{same}_{i1} + \alpha_i + u_{i1}\)

Year 2

\(\text{grades}_{i2} = \alpha + \beta\;\text{same}_{i2} + \alpha_i + u_{i2}\)

First-differencing

\(\Delta\text{grades}_i = \beta\;\Delta\text{same}_i + \Delta u_i\)

• Critical: students must change roommates (\(\Delta\text{same}_i \neq 0\) for some)
• Exogenous mobility design — plausible if the university reassigns rooms

Random Assignment Plus Panel Data Strengthens Identification

You Get What You Pay For

• Random assignment of roommates:
  • \(\beta\) is identified even in cross-section
  • Panel adds precision: removes \(\alpha_i\) from the error \(\to\) smaller variance \(\to\) smaller standard errors
• Potential selection: panel data alone cannot solve the selection problem

Serial Correlation Affects Inference

Clustering Is a Conservative Fix

Clustering does not change \(\hat{\beta}\) — only the standard errors and inference.

Measuring the Effect of Increased Force

All Municipalities Treated Simultaneously

A Linear Time Trend Restores Identification

Parametric but Estimable

\[\text{drug usage}_{it} = \mu\text{post}_t + \theta_i + \gamma t + e_{it}\]

When treatment varies only at the time level, time FE absorb it completely.

When We Do Not Know the Functional Form, Use Dummies

Nonparametric Estimation

\[ \log(\text{earnings})_{it} = \alpha_i + \theta_t + \sum_{j=17}^{85} \gamma_j\;\mathbb{1}[\text{age}_{it} = j] + e_{it} \]

• \(\alpha_i\): individual FE (absorbs ability, education, etc.)
• \(\theta_t\): time FE (absorbs aggregate macroeconomic trends)
• \(\gamma_j\): average difference in log-earnings between workers aged \(j\) and workers aged 16, holding constant \(\alpha_i\) and \(\theta_t\)
  • Nonparametric: no assumption on the shape of the age-earnings relation
  • Parametric: quadratic requires \(f(\text{age}) = \beta_1\;\text{age} + \beta_2\;\text{age}^2\)

Why log?

Imprecision at Extremes Reflects Thin Data

The Variance Formula for Dummy Variables

Each age dummy \(d_j = \mathbb{1}[\text{age}_{it} = j]\) is a binary variable with proportion \(p_j = n_j/n\):

\[ \text{Var}(\hat{\gamma}_j) \propto \frac{\sigma^2}{n \cdot p_j(1 - p_j)} \]

• Most workers aged 25-64 \(\Rightarrow\) \(p_j\) near its maximum \(\Rightarrow\) \(\text{Var}(\hat{\gamma}_j)\) small
• Few workers at ages 16-24 and 65-85 \(\Rightarrow\) \(p_j \approx 0\) \(\Rightarrow\) \(\text{Var}(\hat{\gamma}_j)\) large
• \(\hat{\gamma}_j\) at extreme ages has wide confidence intervals

Nonparametric flexibility comes at the cost of imprecision where data is thin.

Performance Pay Increases Firm-Level Productivity by 20%

But How Much Is Incentives vs Selection?

A firm introduces performance pay. The panel is unbalanced: some workers leave (exiters), some stay (stayers), some join (entrants).

\[ \log(\widehat{\text{productivity}})_{it} = \hat{\alpha}_i + \hat{\beta}\;\text{performance pay}_t \]

The Decomposition Requires No Correlated Shocks

OLS Estimates a Different Quantity, Not a Biased One

• Assumes no other changes coincided with performance pay introduction
  • e.g., new management, different product mix, macroeconomic shifts
• \(\hat{\beta}_{\text{FE}}\) is causal if: \(\mathbb{E}[u_{it} \mid \text{performance pay}_t, \alpha_i] = 0\)
  • Strict exogeneity conditional on individual FE
• The unbalanced panel is central, not incidental
  • Composition effect detected precisely because workers change

Causal interpretation requires that no time-varying confounders coincide with the policy change.

Time-Varying Controls Address Observable Confounders

But Worker Quality Is Unobserved

\[ \text{productivity}_{it} = \beta_1\;\text{contingent}_t + \beta_2\;\text{weather}_t + \beta_3\;\text{width}_{it} + \beta_4\;\text{height}_{it} + e_{it} \]

• Every worker is assigned to different zones each day, so we can control for plant-level conditions (weather, plant size).
• Worker ability differs across periods
  • Best workers leave for blueberry harvest in August (second half)
  • Composition effect is negative: remaining workers are less productive
  • \(\text{contingent}_t\) is collinear with day dummies

Adding Worker Fixed Effects Controls for Unobserved Ability

Controls Handle Observables; FE Handles Unobservables

\[\begin{align*} \text{productivity}_{it} &= \beta_1\;\text{contingent}_t + \beta_2\;\text{weather}_t + \beta_3\;\text{width}_{it} + \beta_4\;\text{height}_{it} \\ &+ \gamma_i + e_{it} \end{align*}\]

• \(\gamma_i\): worker fixed effect — absorbs time-invariant ability
  • Same worker’s productivity compared across wage regimes
• Controls address observable time-varying confounders (weather, plants)
• FE addresses unobservable worker-level confounders (ability, motivation)
• However, only workers observed in both periods contribute to identification
  • If many workers churn, effective sample shrinks

Observable controls and fixed effects address different sources of bias.

Call Centre Productivity Varies Across Operators, Leaders, and Days

Three Sources of Variation

Three-Way Fixed Effects Isolate Leader Quality

Rotation Is the Identification Condition

\[ \text{productivity}_{ijt} = \lambda_i + \mu_j + \theta_t + e_{ijt} \]

• \(\lambda_i\): operator FE — absorbs intrinsic worker ability
• \(\mu_j\): leader FE — the object of interest
• \(\theta_t\): day FE — absorbs common daily shocks
• Identification requires rotation: same operator observed under different leaders
  • Without rotation, \(\lambda_i\) and \(\mu_j\) are not separately identified

The F-Test Provides Evidence for Leader Quality Differences

Joint Significance of Leader Fixed Effects

Multiple fixed effects require sufficient rotation across dimensions for identification.

First-Difference Derivation for General Panel Model

Write the model for \(t = 1\) and \(t = 2\):

\[ \begin{align*} y_{i1} &= \beta_0 + \beta_1 x_{i1,1} + \cdots + \beta_k x_{i1,k} + a_i + v_{i1} \\ y_{i2} &= (\beta_0 + \delta) + \beta_1 x_{i2,1} + \cdots + \beta_k x_{i2,k} + a_i + v_{i2} \end{align*} \]

Subtract: \(a_i - a_i = 0\).

\[ \Delta y_i = \delta + \beta_1\Delta x_{i1} + \beta_2\Delta x_{i2} + \cdots + \beta_k\Delta x_{ik} + \Delta v_i \]

The time-invariant component \(a_i\) has been eliminated. OLS on this equation yields consistent estimates under \(\mathbb{E}[\Delta v_i \mid \Delta x_{i1}, \ldots, \Delta x_{ik}] = 0\).

Return to main

Attenuation Factor: Cross-Section vs First-Differences

Cross-section

\[ \text{plim}\hat\beta_{\text{CS}} = \beta \cdot \frac{\text{Var}(\text{educ}^*)}{\text{Var}(\text{educ}^*) + \text{Var}(e)} \]

First-differences (assuming uncorrelated measurement errors)

\[ \text{plim}\;\hat\beta_{\text{FD}} = \beta \cdot \frac{\text{Var}(\Delta\text{educ}^*)}{\text{Var}(\Delta\text{educ}^*) + \text{Var}(e_{i1}) + \text{Var}(e_{i2})} \]

• Since education changes little over time:
  • \(\text{Var}(\Delta\text{educ}^*) \ll \text{Var}(\text{educ}^*)\).
  • FD \(<\) CS attenuation ratio, and the bias towards zero is worse in FD

Return to main

Why Log?

Three Reasons for Using Log-Earnings

1. Skewness: raw earnings are right-skewed
   • \(\log(\text{earnings})\) yields a more symmetric distribution, closer to normality (supporting AS7)
2. Multiplicative relationships: if earnings = base \(\times\) skill premium \(\times\) experience premium, then
   • \(\log(\text{earnings}) = \log(\text{base}) + \log(\text{skill premium}) + \log(\text{experience premium})\)
   • Log linearises multiplicative structures into additive ones
3. Growth rates: \(\Delta\log(\text{earnings}) \approx \%\Delta\text{earnings}\)
   • Differences in logs approximate percentage changes
   • The natural unit for comparing workers across different baseline earnings

Return to Q9

Age-Period-Cohort Identification Problem

Why Age, Time, and Individual FE Cannot All Be Included

\[ \log(\text{earnings})_{it} = \alpha_i + \theta_t + \sum_{j=17}^{85} \gamma_j\;\mathbb{1}[\text{age}_{it} = j] + e_{it} \]

suffers from a fundamental identification problem:

\[ \text{age}_{it} = \text{year}_t - \text{birth year}_i \]

• \(\text{birth year}_i\) is a linear function of \(\alpha_i\) (time-invariant)
• \(\text{year}_t\) is captured by \(\theta_t\)
• Therefore age is perfectly collinear with \(\alpha_i\) and \(\theta_t\)

Return to Q9

OLS vs FE Decomposition — Formal

Stayers (\(S\)), Exiters (\(X\)), Entrants (\(N\))

\[\begin{align*} \hat{\beta}_{\text{OLS}} &= \bar{y}_{\text{post}} - \bar{y}_{\text{pre}} \\ &= \left[\frac{|S|}{|S|+|N|}\bar{y}^S_{\text{post}} + \frac{|N|}{|S|+|N|}\bar{y}^N_{\text{post}}\right] \\ &\quad - \left[\frac{|S|}{|S|+|X|}\bar{y}^S_{\text{pre}} + \frac{|X|}{|S|+|X|}\bar{y}^X_{\text{pre}}\right] \end{align*}\]

• \(\hat{\beta}_{\text{OLS}}\): weighted across all workers (stayers + exiters + entrants)
• \(\hat{\beta}_{\text{FE}} = \frac{1}{|S|}\sum_{i \in S}(y_{i,\text{post}} - y_{i,\text{pre}})\): averages within stayers only; \(\hat{\beta}_{\text{OLS}} - \hat{\beta}_{\text{FE}}\) captures composition

Return to Q10

AKM: Matched Employer-Employee Framework

Abowd, Kramarz, and Margolis (1999, Econometrica)

The call centre model (Q12) is a simplified version of AKM’s framework. In AKM:

\[ \log(\text{wages})_{it} = \alpha_i + \psi_{J(i,t)} + x'_{it}\beta + e_{it} \]

• \(\alpha_i\): worker fixed effect (ability, human capital)
• \(\psi_{J(i,t)}\): firm fixed effect for the firm \(J\) employing worker \(i\) at time \(t\)
• Identification requires worker mobility across firms (exogenous mobility assumption)
  • Same logic as Q12: operators rotate across teams
  • Without mobility, \(\alpha_i\) and \(\psi_{J(i,t)}\) are not separately identified

Return to Q12

Perfect Collinearity: Day FE and Post

Formal Proof That \(\mu\) Is Not Identified

With day dummies \(d_1, \ldots, d_T\) and treatment at day \(k\):

\[ \text{post}_t = \sum_{s=k}^{T} \mathbb{1}[t = s] = d_k + d_{k+1} + \cdots + d_T \]

• \(\text{post}_t\) is an exact linear combination of day dummies
• With \(d_1\) as reference: \(T-1\) dummies plus \(\text{post}_t\) give \(T\) variables with \(T-1\) independent columns
• The design matrix \(X'X\) is rank-deficient by 1 \(\Rightarrow\) \(\mu\) not identified
• Resolution: replace day dummies with a parametric trend (\(\gamma\;\text{time}_t\))

Return to Q8